Register to reply

How to use Euler's angle theorem in rotation of a coordinate

Share this thread:
Dec4-12, 07:49 PM
P: 9
If i have a point at (0,0,5) in x,y,z system, then i make 2 rotation on the point with center at origin.
i)the first rotation is on y-axis with angle P in clockwise direction.
ii) the second rotation is on the point's new x-axis rotate in angle Q in clockwise direction.

How can i find the new coordinates after these 2 rotation with Euler's rotation theorem?
Phys.Org News Partner Science news on
'Office life' of bacteria may be their weak spot
Lunar explorers will walk at higher speeds than thought
Philips introduces BlueTouch, PulseRelief control for pain relief
Dec4-12, 10:38 PM
Sci Advisor
HW Helper
P: 9,930
That theorem only tells you the net transformation is a rotation. Rodrigues' rotation formula might be more use.
Dec5-12, 02:57 AM
P: 9
using the point (0,0,5) , i make a rotation of 30 degree in clockwise direction about y axis,

so i should get the rotation matrix of

| cos 30 0 -sin 30 | |x| | cos 30 0 sin 30 | |x|
| 0 1 0 | |y| or | 0 1 0 | |y|
| sin 30 0 cos 30| |z| | -sin 30 0 cos 30| |z|

which one is the right one or i got both wrong?

Dec5-12, 05:59 AM
Sci Advisor
HW Helper
BobG's Avatar
P: 2,284
How to use Euler's angle theorem in rotation of a coordinate

Second one is correct (I think, they don't line up very well).

One rotation matrix for positive (direction of a right hand screw, or counter-clockwise as viewed from the outside) rotations:

1        0        0
0     cos x     sin x
0    -sin x     cos x
With columns x, y, z and rows x, y, z, shift the matrix so the 1 lies at the intersection of whichever axis you're rotating around.

For negative (or clockwise), reverse the sign of the sines.

Since you're dealing with orthogonal matrices, the safer way is that clockwise is the inverse of counter-clockwise and the inverse of an orthogonal right hand matrix (which your rotation matrices are) is the transpose. In other words, take the transpose of the counter-clockwise matrix to get the clockwise rotation matrix.

The latter helps you when you start combining rotations into a single matrix and you don't want to have to reaccomplish the work for the opposite direction.

Additionally, sometimes the known angle won't be the rotation angle, but the complement of the rotation angle. Instead of building an ugly rotation matrix consisting of cos (90 - x), sine will substituted for cosine and cosine for sine to make a cleaner matrix consisting of just x.
Dec6-12, 02:11 AM
P: 9
lets say if i have a Z-axis which perpendicularly point into the screen, X-axis which point to its right side and Y-axis which point upward and a point with coordinate ( 0 , 0 , 5 ).

then i rotate the point's y-axis 30 degree in clockwise direction.
so the rotation operation matrix should be

|cos 30      0    sin 30   |
|    0         1      0    |
|-sin 30      0    cos 30  |
Sorry about the disoriented matrix in previous post...

then on the point's new x-axis, i rotate the point 60 degree counter clockwise, so the point will move upward. And the matrix is:

|    1        0        0       |
|    0       cos 60   sin 60   |
|    0      -sin 60   cos 60   |
so to get the point final position, i just multiply all the stuff together like:

|cos 30      0    sin 30   | |    1        0        0       | | 0 |     |1.25   |
|    0        1     0      | |    0       cos 60   sin 60   | | 0 | =   |4.3301 |
|-sin 30      0    cos 30  | |    0      -sin 60   cos 60   | | 5 |     |2.165  |

Is that the right way?
Dec6-12, 07:05 PM
P: 9
What is the A=BCD mean in this page, and how do they get the value of a11 , a12 , a13...a33, i thought it is just multiplication of matrices
Dec7-12, 01:37 AM
P: 9
I get it now, thanks everyone for teaching me.

Register to reply

Related Discussions
Need to understand: derivating euler rotation from fixed axis rotation Engineering, Comp Sci, & Technology Homework 1
Solving for Euler angles and 3-D coordinate Rotations. Advanced Physics Homework 2
Showing that the euler lagrange equations are coordinate independent Classical Physics 2
Rotation of coordinate axes Calculus & Beyond Homework 0
Coordinate transformation under rotation General Physics 5